Let P and Q be Hermitian positive definite matrices. We prove that x*Px < or eq. x*Qx, for all x in C^n (C : complex numbers) if and only if x*Q^-1 x < or eq. x*P^-1 x for all x in C^n. I guess I should use the definition of a hermitian positive definite matrix being x*Px > 0 , for all x in C^n but I am not sure how to proceed to get both the P and Q in the inequality. Should I try and multiply both sides of the inequality by x and x*?